The problem is that it is not a worry that is reasonable if HH was not Town reading Cheese.
The way it was worded makes it seem like they don’t think Cheese will flip Mafia, but didn’t want to town read them.
That to me seems way more likely that HH is Mafia worried that if Cheese who they know is Town gets shot that Cheese will guarantee kill a Mafia, HH.
From a Town HH PoV the fear that Cheese will flip Town doesn’t seem worrysome since even if we assume HH was right that Cheese would instantly shoot HH, HH would have control over the shot and not die. So unless Town HH has anxiety about leading the Town, I don’t see the justification for the fear. At least not moreso than Mafia worrying they’d be shot.